As shown in [1], the category OrdGrp of preordered groups and monotone group homomorphisms does not satisfy the Split Short Five Lemma, although the behaviour of split extensions in OrdGrp has some interesting facets.
In this talk we will study these split extensions, focusing in particular in relative protomodularity and action representability.
This talk is based on joint work with Nelson Martins-Ferreira and Andrea Montoli [1], and Carla Ruivo Morouço [2].

References:
[1] M.M. Clementino, N. Martins-Ferreira, A. Montoli, On the categorical behaviour of preordered groups, J. Pure Appl. Algebra 223 (2019), 4226-4245.
[2] M.M. Clementino, C. Ruivo Morouço, On split extensions of preordered groups, in preparation.